[#P3046] Consecutive Primes in Arithmetic Progression

Problem. Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?

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Definitions and notation

1Context

This problem concerns the existence of arbitrarily long sequences of consecutive primes that form arithmetic progressions. The question asks whether for every length \(k \geq 3\), there exists some starting prime such that the \(k\) consecutive primes beginning at that point are equally spaced.

2Definitions

Definition 1 (A set \(s\) of natural numbers). A set \(s\) of natural numbers is an arithmetic progression of length \(l\) if there exist natural numbers \(a\) (the first term) and \(d\) (the common difference) such that \(s = \{a + nd \mid n \in \mathbb{N}, n < l\}\).

Definition 2 (A set \(s\) of natural numbers). A set \(s\) of natural numbers is a prime progression of length \(l\) if there exists a natural number \(a\) such that \(s\) has cardinality \(l\) and equals the set of the first \(l\) consecutive primes starting from the \(a\)-th prime, i.e., \(s = \{p_{a+n} \mid n \in \mathbb{N}, n < l\}\) where \(p_m\) denotes the \(m\)-th prime number.

Definition 3 (A set \(s\). A set \(s\) is an arithmetic progression of consecutive primes of length \(k\) if it is both an arithmetic progression of length \(k\) and a prime progression of length \(k\).

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?^[1] Relevance to this problem For erdos problem 141, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes?^[1] Relevance to this problem For Consecutive Primes in Arithmetic Progression, record erdos-problem-141-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 141: For every integer \(k \geq 3\), does there exist a set of \(k\) consecutive primes in arithmetic progression? That is, for each \(k \geq 3\), does there exist a set \(s\) of natural numbers such that \(s\) forms an arithmetic progression of length \(k\) and consists of \(k\) consecutive primes? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 141 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 141 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemConsecutive Primes in Arithmetic Progression

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

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Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“Consecutive Primes in Arithmetic Progression.” TheoremDB. P3046. Problem statement; statement text SHA-256 020be404a45fe53b742cb0c6b093f78a8f36a5fb8d66091679ff0243122276c8. https://theoremdb.org/statement/?ref=P3046

BibTeX

@misc{theoremdb-problem-020be404a45fe53b742cb0c6b093f78a8f36a5fb8d66091679ff0243122276c8,
  title = {{Consecutive Primes in Arithmetic Progression}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 020be404a45fe53b742cb0c6b093f78a8f36a5fb8d66091679ff0243122276c8},
  url = {https://theoremdb.org/statement/?ref=P3046}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 141, maintained status record. Erdős Problems record 141, checked 2026-08-01. Problem 141; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 141; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Consecutive Primes in Arithmetic Progression: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 141 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Consecutive Primes in Arithmetic Progression: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 141. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/141.lean:L75; theorem erdos_141; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Consecutive Primes in Arithmetic Progression: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P3046] Consecutive Primes in Arithmetic Progression

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3What counts as a solution

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Background

Definitions

What counts as a solution

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